Theorem newssno

Description: New sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024)

	Ref	Expression
Assertion	newssno	$$⊢N\left(A\right)\subseteq \mathrm{No}$$

Proof

Step	Hyp	Ref	Expression
1		newf	$$⊢N:\mathrm{On}⟶𝒫\mathrm{No}$$
2		0elpw	$$⊢\varnothing \in 𝒫\mathrm{No}$$
3	1 2	f0cli	$$⊢N\left(A\right)\in 𝒫\mathrm{No}$$
4		elpwi	$$⊢N\left(A\right)\in 𝒫\mathrm{No}\to N\left(A\right)\subseteq \mathrm{No}$$
5	3 4	ax-mp	$$⊢N\left(A\right)\subseteq \mathrm{No}$$